Dynamics of Microtubule Growth & Catastrophe

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1 Dynamics of Microtubule Growth & Catastrophe T. Antal, P. Krapivsky, SR, M. Mailman, B. Chakraborty, q-bio/ U Mass Amherst, March 15, 2007 Basic question: Why do microtubules fluctuate wildly under steady conditions? Main result: These large fluctuations & rich dynamics are captured by a simple, soluble model. Outline: What is a microtubule? What does it do? Microscopic modeling. Master equation & probabilistic results.

2 What is a Microtubule? a Howard & Hyman, Nature 2003 micrograph of a stained cell centrosome b schematic picture of a cell microtubules

3 What Do Microtubules Do? a chromosome transport during mitosis metaphase b spindle c spindle transport into bud

4 What is a Microtubule Made Of? from wikipedia

ubules are long polymers of the protein w n thatarticles form a network within cells to based on changes in the arrangement of tubulin subunits in the polymer lattice.

5 ubules are long polymers of the protein w n thatarticles form a network within cells to based on changes in the arrangement of tubulin subunits in the polymer lattice. Tubulin molecules in microtubules are arranged in 13 lines called protofilaments, which lie parallel to the microtubule axis. When microtubules depolymerize,cthese protofilaments curve outd GDPand in the presence of microtubulewards, associated proteins or certain divalent cations, " they bend back on themselves to form stable rings of GDP-tubulin (Fig. 1a)6. GTP hydrolysis was proposed to destabilize microtubules, and drive dynamic instability, by promoting outward curving5, although the mechanism coupling hydrolysis to curving wasdocking unknown. By comparing the structure of GDPprotofilament rings1 with that of micro7 tubules, Wang and Nogales1 reveal how GTP Hydrolysis hydrolysis promotes protofilament curving and thus destabilizes the microtubule lattice. The GDP-protofilament is bent at both the inter- and intra-dimer interfaces, making it curve outwards from the microtubule. Within conformation. Simple dynamic instability theory predicts that the preferred conformation of GTP-tubulin should be that of the microtubule lattice; that is, straight protofilaments. But in the GMPCPP structure they in D fact curve outwards, albeit to a lesser extent than GDP protofilaments. This structure required cooling, which induces a conformational change in tubulin, so the geometry might differ from anything that occurs normally. With that caveat, the combined data support a two-step model for microtubule growth, with initial polymerization into gently curved sheets, followed by tube closure1,5. Exactly when GTP hydrolysis occurs is not clear. GTP analogue protofilaments roll up into microtubules on warming1, showing that hydrolysis isdnot necessary for tube closure. The alternative forms of the GTP-tubulin lattice probably have similar energies, and may interconvert at growing ends, while maintaining a GTP cap (Fig. 1a). A greater understanding of how tubulin How Do Microtubules Grow and Shrink? range the cell components and provide ort tracks for motor proteins. Rather hydrolysis eing staticcycle permanent structures, microthe sincontinuously grow and shrink through a of the tubulin olymerization and depolymerization! rotofilament. ulin. Such processes are central to the GTP the microtubule ubules spatial organization and their gto!-subunit generate the forces necessary to functo theissue, lattice- Wang and Nogales (page nndthis b ubules athigh-resolution eport structures of two protofilaments ative polymeric states of tubulin, which ends curl. The e insights into the molecular mechahown in panel d. that power growth and shrinkage. e a straight ulin is a stable dimer of " and! subthe straight wall both of which bind guanine nucleogtp induces a Guanosine triphosphate (GTP) that is d is constrained to!-tubulin is hydrolysed to guanosine places stress on sphate (GDP) during microtubule ng bly, and this rotofilament to nucleotide regulates tubunformation and behaviour, with GTP ing polymerization, and GDP depolyation. In the presence of tubulin and ndividual microtubule ends tend to or many micrometres, and then switch rtening. This transition, called catae ends and destabilize them (Fig.a5a), and 48 e, occurs spontaneously with pure oteins (or +TIPs ), which bindtubuto the dtubule constant levels, although cells andgtp at least in some casesinstabilize gulated other proteins. growthby phase (Fig. 5c). The resulting 2 mic instability allows microtubule ends ciently explore their surroundings3 and d-binding proteinswork are by thepushing MCAKs,and also orm mechanical 51,52 4 kinesins thanchemimoving ge.unusual A central question,israther how the rotubules like other motoris proteins, ergy from GTP hydrolysis harnesseduse ysis to bind to the of microtubules, er both growth andends shrinkage of micross in and thus trigger depolymerization53,54. dynamic instability. MCAK (XKCM1) from egg extracts dramatal models emphasized a thermody55 f the microtubule arrays by suppressing microtubules grow in the presence of GFP CLIP-170, bright patches can be seen at the growing end; these patches then disappear when the microtubule stops growing63,64 (Fig. 5c). Both the S. pombe65 and the & S. cerevisiae66 homologues of CLIP-170 have also been shown Mahadevan to target microtubule ends. Work in tissue culture cells illustrates the Nature 2005 interaction between CLIP-170 and dynamic microtubules. Here, microtubules growing from centrosomes initially exhibit similar dynamic instability properties as described in vitro67. That is, they have a low catastrophe rate and if a microtubule does catastrophe, it usually shrinks back to the nucleation centre because the rescue rate is also low. But when a microtubule reaches the cell periphery, the stability of its plus end changes markedly. Here, microtubules that undergo catastrophe rapidly rescue, and microtubules close to the Mitchison,

6 Microtubule Evolution Fygenson et al., PRE (1994) attachment detachment

7 Microscopic Model 1. Growth: attachment of a GTP + monomer. + = + + rate λ = + rate pλ 2. Conversion: GTP + hydrolysis to a GDP monomer. + = rate 1 3. Shrinking: detachment of a GDP from the microtubule end. = rate µ

8 Cartoon of Growing Microtubule no detachment, µ = 0 islands <L>=! t + GTP tail GTP + populated zone cap +

9 Cartoon of Shrinking Microtubule λ < µ convert shrink

10 Dynamics of Unrestricted Growth no detachment: µ = 0 color-blind attachment: p = 1 N number of GTP + Rate equation: d dt N = λ N N = λ Π N prob. that tubule contains N GTP + Master equation: Solution: dπ N dt conversion: N N-1 attachment: N N+1 = (N +λ)π N +λπ N 1 +(N +1)Π N+1 Π N (t) = [λ(1 e t )] N N! attachment: N-1 N e λ(1 e t ) conversion: N+1 N

11 Generating Function Solution generating function: Π(z) N=0 Π Nz N dπ N dt = (N +λ)π N +λπ N 1 +(N +1)Π N+1 Π t = (1 z) ( Π z λπ ). define Q = Πe λz, y = log(1 z), Q t + Q y = 0, formal solution: Q = F (t y) = F (e t (1 z)) for initial condition Π N (t = 0) = δ N,0 Π(z, t) = e λ(1 z)(1 e t )

12 Cap Length Distribution n k probability that cap length equals k! <L>= t cap Master equation: Steady state: ṅ k = λ(n k 1 n k ) kn k + N k = λ attach to (k-1)-cap k + λ N k 1 attach to k-cap convert k-cap N k = s k convert >k-cap s k+1 n s k n s Solution: N k = λk Γ(1 + λ) Γ(k λ) k = k 0 kn k πλ 2 λ

13 Island Distributions: Continuum Limit λ prob. GTP + distance x from tip does not convert: e τ = e x/λ extreme criterion: 1 GTP + beyond l 1 = x l e x/λ = (1 e 1/λ ) 1 e l/λ prob. GTP x <L>=! t The GTP + populated zone: l = λ ln λ

14 Cap Length Distribution (continuum) prob. that cap has length k: (1 e (k+1)/λ ) monomer k+1 converts k j=1 e j/λ all monomers within k of the tip do not convert <L>=! t n k k + 1 λ e k(k+1)/2λ k k λ

15 Island Length Distributions prob. positive island at x has length k: (1 e x/λ )(1 e (x+k+1)/λ ) monomer at x converts monomer at x+k+1 converts k j=1 e (x+j)/λ k monomers in between do not convert (1 e x/λ ) 2 e kx/λ prob. GTP <L>=! t x I k density of islands of length k = 0 dx (1 e x/λ ) 2 e kx/λ = k 2λ k(k+1)(k+2)

16 prob. negative island at x has length k: e 2x/λ (1 e x/λ ) k end monomers do not convert k monomers in between convert <L>= x! t k J k = 0 dx e 2x/λ (1 e x/λ ) k = λ (k + 1)(k + 2) away from tip { shorter GTP + islands: longer GDP islands: k 3 distribution k 2 distribution

17 Catastrophes convert shrink to zero

18 catastrophe probability: C(λ) = λ (1 e n/λ ) n=1 prob. last monomer converts before new attachment prob. rest of tubule has converted Dedekind η function: η(z) = e iπz/12 n=1 (1 e 2πinz ) η( 1/z) = iz η(z) (1 e 2an ) = n=1 π a e(a b)/12 (1 e 2bn ) n=1 b = π2 a C(λ) = 2πλ λ e π λ/6 e 1/24λ (1 e 4π2λn ) n 1 2π 2 λ e π λ/6

19 Avalanches convert shrink

20 avalanche probability: A k = λ k 1 n=1 prob. last monomer converts before new attachment (1 e n/λ ) prob. k monomers of tubule have converted expand exponential to 2nd order: A k = λ k Γ(k) k 1 ( 1 n 2λ ) λ k Γ(k) e k2 /4λ n=1

21 Microtubule Phase Diagram λ, µ 1 fast conversion end is GDP attach (grow) with rate pλ v = pλ µ µ = pλ detach (shrink) with rate µ λ, µ 1 during growth: L = vt e π2 λ/6 rescue occurs if: t shrink = L µ > 1 pλ ignore power-law terms in λ compared to exponential terms µ p e π2 λ/6

22 Microtubule Phase Diagram 8 6 (a) p=1 compact µ = pλ 2 nd order approx. µ 4 data 2 0 growing ! 8 6 (b) p=0.1 µ compact growing !

23 Summary & Outlook microtubules have rich dynamical behavior dynamics solvable by probabilistic approaches reality checks comparison with real data validation of model parameters

Re-entrant transition in a one-dimensional growth model

Re-entrant transition in a one-dimensional growth model Ostap Hryniv Dept. of Mathematical Sciences Durham University Ostap.Hryniv@durham.ac.uk X International Conference Stochastic and Analytic Methods